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Wize University Physics Textbook (Master) > Thermodynamics

Heat Engines

Thermodynamic CyclesPrevious Section
Refrigerators and Heat PumpsNext Section
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Heat Engines & Refrigerators

Heat Engine: Any device that transforms heat partly into the mechanical energy or work is called a heat engine.
  • The net heat given into the system is called QHQ_H which is positive
  • Part of QHQ_H is transformed to work and the rest, QCQ_C, is leaving the system.
  • QCQ_C is negative
  • According to Conservation of Energy Law: QH=W+∣QC∣Q_H=W+|Q_C |
  • The efficiency of a heat engine is defined as: e=WQHe=\dfrac{W}{Q_H }
  • The less energy given into the system and the more part of the heat transferred to work, the better heat engine!
e=WQH=QH−∣QC∣QH=1−∣QC∣QH=1+QCQHe=\dfrac{W}{Q_H} =\dfrac{Q_H-|Q_C |}{Q_H} =1-\dfrac{|Q_C |}{Q_H} =1+\dfrac{Q_C}{Q_H}
  • ee is always smaller than 1 which means that efficiency is always less than 100%
  • Heat engine cycle is always clockwise
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Refrigerator: A device which transfers heat from a cold place to a hot place. In contrast to heat engines, refrigerator requires a net input of mechanical energy or work. So Wt<0W_t<0
  • The heat which is taken from the cold place is called QCQ_C and is positive
  • The heat which is given off to the hot place is called QHQ_H and is negative. Similar to heat engine, for any refrigerator we have:
∣QH∣=QC+∣W∣|Q_H |=Q_C+|W|
  • The coefficient of performance for a refrigerator is defined as:
K=QC∣W∣=QC∣QH∣−QC=∣QH∣∣W∣−1K= \dfrac{Q_C}{|W|} =\dfrac{Q_C}{|Q_H |-Q_C}= \dfrac{|Q_H |}{|W|} -1

  • The more heat taken from the cold place and the less net input work required, the better the refrigerator!
  • A refrigerator cycle is always counterclockwise.

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Example: The Carnot Cycle

A Carnot engine is operated between THT_H and TCT_C according to the PV – Diagram on the right where the processes a→ba\to b and c→dc\to d are isothermal and the processes b→cb\to c and d→ad\to a are adiabatic.

Find the efficiency of this engine as a function of TCT_C and THT_H
Qbc=Qda=0Q_{bc}=Q_{da}=0 adiabatic
Qab:ΔT=0→Wab=Qab>0Q_{ab}:\Delta T=0\to W_{ab}=Q_{ab}>0
Qab=nRTHln⁡(VbVa)Q_{ab}=nRT_H\ln\bigg(\dfrac{V_b}{V_a}\bigg)
Qcd=Wcd<0→Qcd=nRTCln⁡(VdVc)Q_{cd}=W_{cd}<0\to Q_{cd}=nRT_C\ln\bigg(\dfrac{V_d}{V_c}\bigg)
e=1+QcQH=1+TcTHln⁡(VdVC)ln⁡(VbVa)e=1+\dfrac{Q_c}{Q_H}=1+\dfrac{T_c}{T_H}\dfrac{\ln\Big(\frac{Vd}{V_C}\Big)}{\ln\Big(\frac{Vb}{V_a}\Big)}
TVγ−1TV^{\gamma-1}=Constant. For adiabatic processes
For b→cb\to c: THVbγ−1=TCVcγ−1→THTC=(VcVb)γ−1T_HV_b^{\gamma-1}=T_CV_c^{\gamma-1}\to \dfrac{T_H}{T_C}=\bigg(\dfrac{V_c}{V_b}\bigg)^{\gamma-1}
for d→ad\to a: TCVdγ−1=THVaγ−1→THTC=(VdVa)γ−1T_CV_d^{\gamma-1}=T_HV_a^{\gamma-1}\to \dfrac{T_H}{T_C}=\bigg(\dfrac{V_d}{V_a}\bigg)^{\gamma-1}
→VcVb=VdVa\to\dfrac{V_c}{V_b}=\dfrac{V_d}{V_a} or VdVc=VaVb\dfrac{V_d}{V_c}=\dfrac{V_a}{V_b} (plug into efficiency equation)
e=1−TCTHe=1-\dfrac{T_C}{T_H}
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Otto Cycle is made of two isochoric processes at 𝑉0 and 𝑟𝑉0 for 𝑟 > 1 and two adiabatic processes which compress and extend the gas between these two volumes.
Plot PV curve of this cycle and find its efficiency in terms of only compression ratio, 𝑟 and 𝛾.
Extra Practice
A Carnot engine has an efficiency of 33.0% It operates between constant reservoirs differing in temperature by 95oC.
What are the temperatures of the two reservoirs?
Entropy: Heat Engines
A cylinder contains Nitrogen gas (𝐶v = 20.8Jmol.K\frac{J}{mol.K} , 𝛾 = 1.4) at a pressure of 3 atm and a temperature of 250K. Its initial volume is 2.0 liters. The Nitrogen gas is next carried through the following processes:
a. It is heated in an isochoric process to a temperature of 375K
b. It is expanded at a constant temperature to a volume of 3.0 liters
Heat Engines: Otto Cycle
Otto Cycle is made of two isochoric processes at 𝑉0 and 𝑟𝑉0 for 𝑟 > 1 and two adiabatic processes which compress and extend the gas between these two volumes.
Plot PV curve of this cycle and find its efficiency in terms of only compression ratio, 𝑟 and 𝛾.
In a Stirling heat engine, a monatomic ideal gas at initial volume V1 and temperature T1, takes in heat at a constant volume until its temperature is T2, and then expands isothermally until its volume is V2. It gives out heat at a constant volume until its temperature is again T1 and then returns to its initial state by isothermal contraction. Calculate the efficiency and compare with a Carnot engine operating between the same two temperatures.
Thermodynamics: Heat Engines
A diatomic gas undergoes some thermodynamical processes according to the following plot. Imagine the process from a to b could be changed in order to get different efficiency for the following cycle under the condition that the temperature of the gas cannot exceed the temperature at point a during the entire cycle. What is the work done by the gas from a to b in a process which makes the efficiency of the heat engine equal to the half of the efficiency of a Carnot engine which works in the same range of temperatures? (Assume we have 2 moles of the gas)
PV Curve: Heat Engines
In a Diesel cycle Vo volume of gas is expanded in an isobaric process to
𝛼𝑉o and then is expanded more to 𝑟𝑉o in an adiabatic process. The gas in then cooled down via an isochoric process and then is compressed back to its initial volume. Draw PV curve of this cycle and find the efficiency of this cycle only in terms of 𝛼, 𝑟 ,and 𝛾.
Entropy: Heat Engines
A cylinder contains a diatomic gas at a pressure of 3 atm and a temperature of 250K. Its initial volume is 4.0 liters. The gas is next carried through the following processes:
a. It is expanded isobarically to a temperature of 375K.
b. It is cooled down to 250K in an isochoric process.
Which one of the following statements best describes the operation of a heat engine?
For heat engine cycle is
clockwise
while for heat pump cycle is
counterclockwise
An engine is found to have an efficiency of 0.40. If it does 200 J of work per cycle, what are the corresponding quantities of heat absorbed and rejected?